ABCD is a quadrilateral in which P, Q, R and S are mid-points of the sides AB, BC, CD and DA (see Fig 8.29). AC is a diagonal. Show that:(i) SR || AC and SR = 1/2 AC(ii) PQ = SR(iii) PQRS is a parallelogram.

To prove the given conditions, we will use the Mid-point Theorem which states that the line segment joining the midpoints of any two sides of a triangle is parallel to the third side and equal to half of it.

Given: Quadrilateral ABCD with AC as a diagonal. P, Q, R and S are mid-points of the sides AB, BC, CD and DA respectively.

(i) SR || AC and SR = 1/2 AC

In triangle ADC, S and R are the mid-points of AD and DC respectively. By the Mid-point theorem, SR is parallel to AC and SR is half of AC.

(ii) PQ = SR

In triangle ABC, P and Q are the mid-points of AB and BC respectively. By the Mid-point theorem, PQ is parallel to AC and PQ is half of AC. But from (i), SR is also half of AC. Therefore, PQ = SR.

(iii) PQRS is a parallelogram

We have already proved that SR is parallel to AC and PQ is parallel to AC. Therefore, SR is parallel to PQ. Also, since P and S are mid-points of AB and AD respectively, PS is parallel to BD. Similarly, QR is parallel to BD. Therefore, PS is parallel to QR. Hence, PQRS is a parallelogram (since opposite sides are parallel).